Make a GitHub Project Page that's an exact mirror of your Master branch

GitHub has a Project Pages feature that creates a webpage based on the content of the gh-pages branch of a repository. I asked, and subsequently answered, the question of how to get the Project Page to be an exact mirror of the Master branch over on Stack Overflow.

One way I found is to run the following command, which sets a configuration setting that causes every subsequent instance of git push to push the Master branch of the repository into the gh-pages branch:
git config --add remote.origin.push +refs/heads/master:refs/heads/gh-pages

The trouble with this solution is that I'm using the GitHub Mac App, which doesn't use a strict git push. According to their Help page, it does a git pull --rebase. Additionally, even when I select Push from the Repository menu, my git config line isn't adhered to. I'm not sure what's up with that...

I ran through a bunch of my own tests, trying to figure out how this branch mirroring can be done in a GitHub-For-Mac-App-Friendly way. It doesn't seem like you need to go through the exact process they've described to create the gh-pages branch. Over there, they provide the following process along with the disclaimer "Caution: make your working directory clean before you do this (either stash or commit), otherwise this will lose any changes you've made to your project since the last commit.":